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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Pentatope number ： ウィキペディア英語版 Pentatope number A pentatope number is a number in the fifth cell of any row of Pascal's triangle starting with the 5-term row 1 4 6 4 1 either from left to right or from right to left. The first few numbers of this kind are : : 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365 Pentatope numbers belong in the class of figurate numbers, which can be represented as regular, discrete geometric patterns. The formula for the ''n''th pentatopic number is: : $=\; \backslash frac\; =\; .$ Two of every three pentatope numbers are also pentagonal numbers. To be precise, the (3''k'' − 2)th pentatope number is always the ((3''k''2 − ''k'')/2)th pentagonal number and the (3''k'' − 1)th pentatope number is always the ((3''k''2 + ''k'')/2)th pentagonal number. The 3''k''th pentatope number is the generalized pentagonal number obtained by taking the negative index −(3''k''2 + ''k'')/2 in the formula for pentagonal numbers. (These expressions always give integers). The infinite sum of the reciprocals of all pentatopal numbers is $4\; \backslash over\; 3$.〔. Theorem 2, p. 435.〕 This can be derived using telescoping series. : $\backslash sum\_^\backslash infty$ Pentatopal numbers can also be represented as the sum of the first n tetrahedral numbers.〔 ==Test for pentatope numbers== :$\backslash frac$ is triangular number. then : $8$ *\frac+1 is perfect square. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Pentatope number」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.